[Proof of theo4]
It follows from the Cauchy-Schwarz inequality [1]} that
\(\begin{alignedat}{-1}&\mathbb {E}[2\langle \Psi ^{k+1}-\Psi ^\star ,\widehat{Y}^{k+1}-Y^k\rangle ]\\&\le \zeta \mathbb {E}\Vert \widetilde{\Psi }^{k+1}\Vert ^2+\zeta ^{-1}\mathbb {E}\Vert \widehat{Y}^{k+1}-Y^k\Vert ^2,\forall \zeta >0.\end{alignedat}\) 
